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On a \(\Phi\)-Kirchhoff multivalued problem with critical growth in an Orlicz-Sobolev space. (English) Zbl 1304.35254

Summary: This paper is concerned with the multiplicity of nontrivial solutions in an Orlicz-Sobolev space for a nonlocal problem with critical growth, involving \(N\)-functions and theory of locally Lipschitz continuous functionals. More precisely, in this paper, we study a result of multiplicity to the following multivalued elliptic problem: \[ \begin{cases} -M \left (\displaystyle \int_\varOmega \varPhi(|\nabla u|){\mathrm {d}}x \right ) \Delta_\varPhi u \in \partial F(\cdot,u)+\alpha h(u) \quad \text{in }\varOmega, \\ u \in W^{1}_0L_\varPhi(\varOmega), \end{cases} \] where \(\varOmega \subset \mathbb{R}^N\) is a bounded smooth domain, \(N \geqslant 3\), \(M\) is a continuous function, \(\varPhi\) is an \(N\)-function, \(h\) is an odd increasing homeomorphism from \(\mathbb{R}\) to \(\mathbb{R}\), \(\alpha\) is positive parameter, \(\Delta_\varPhi\) is the corresponding \(\varPhi\)-Laplacian and \(\partial F(\cdot,t)\) stands for Clarke generalized of a function \(F\) linked with critical growth. We use genus theory to obtain the main result.

MSC:

35J30 Higher-order elliptic equations
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